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Full-Text Articles in Engineering

Kalman Consensus Strategies And Their Application To Cooperative Control, Randal Beard, Derek Kingston, Wei Ren Nov 2004

Kalman Consensus Strategies And Their Application To Cooperative Control, Randal Beard, Derek Kingston, Wei Ren

Faculty Publications

Sponsorship: AFOSR, NSF. In this paper, we propose discrete-time and continuous-time consensus update schemes motivated by the discrete-time and continuous-time Kalman filters. With certainty information encoded into each agent, the proposed consensus schemes explicitly account for relative confidence in the information that is communicated from each agent in the team. We show mild sufficient conditions under which consensus can be achieved using the proposed schemes in the presence of switching interaction topologies. The Kalman consensus scheme is shown to be input-to-state stable. We show how to exploit this fact in multi-agent cooperative control scenarios.


Simulation Of Gate Lag And Current Collapse In Gallium Nitride Field-Effect Transistors, N. Braga, R. Mickevicius, R. Gaska, M. S. Shur, M. Asif Khan, Grigory Simin Nov 2004

Simulation Of Gate Lag And Current Collapse In Gallium Nitride Field-Effect Transistors, N. Braga, R. Mickevicius, R. Gaska, M. S. Shur, M. Asif Khan, Grigory Simin

Faculty Publications

Results of two-dimensional numerical simulations of gate lag and current collapse in GaN heterostructurefield-effect transistors are presented. Simulation results clearly show that current collapse takes place only if an enhanced trapping occurs under the gate edges. Hot electrons play an instrumental role in the collapse mechanism. The simulation results also link the current collapse with electrons spreading into the buffer layer and confirm that a better electron localization (as in a double heterostructurefield-effect transistor) can dramatically reduce current collapse.


A Review Of Antennas And Propagation For Mimo Wireless Communications, Michael A. Jensen, Jon W. Wallace Nov 2004

A Review Of Antennas And Propagation For Mimo Wireless Communications, Michael A. Jensen, Jon W. Wallace

Faculty Publications

Multiple-input-multiple-output (MIMO) wireless systems use multiple antenna elements at transmit and receive to offer improved capacity over single antenna topologies in multipath channels. In such systems, the antenna properties as well as the multipath channel characteristics play a key role in determining communication performance. This paper reviews recent research findings concerning antennas and propagation in MIMO systems. Issues considered include channel capacity computation, channel measurement and modeling approaches, and the impact of antenna element properties and array configuration on system performance. Throughout the discussion, outstanding research questions in these areas are highlighted.


Resolution Scaling In Noncontact Scanning Impedance Imaging, Benjamin C. Green, Tao Shang, Jacey C. Morine, Hongze Liu, Stephen M. Schultz, Travis E. Oliphant, Aaron R. Hawkins Oct 2004

Resolution Scaling In Noncontact Scanning Impedance Imaging, Benjamin C. Green, Tao Shang, Jacey C. Morine, Hongze Liu, Stephen M. Schultz, Travis E. Oliphant, Aaron R. Hawkins

Faculty Publications

Noncontact scanning impedance imaging has been presented as a method to provide high resolution, high contrast images for a variety of material systems. This technique combines electrical impedance measurements with very high resolution scanning. This article reports on efforts to scale this technique down to the very important single micron range and reveals measurements for both thick and thin samples with a measured minimum resolution below 30 µm. A design for a shielded impedance probe applicable to this process is outlined and probes of several different sizes were made and tested. Fabrication of these impedance probes is explained and a …


Integrated Optical Waveguides With Liquid Cores, Aaron R. Hawkins, John P. Barber, Dongliang Yin, D. W. Deamer, Holger Schmidt Oct 2004

Integrated Optical Waveguides With Liquid Cores, Aaron R. Hawkins, John P. Barber, Dongliang Yin, D. W. Deamer, Holger Schmidt

Faculty Publications

We report the design, fabrication, and demonstration of single-mode integrated optical waveguides with liquid cores. The principle of the device is based on antiresonant reflecting optical (ARROW) waveguides with hollow cores. We describe design principles for waveguide loss optimization down to 0.1/cm. Using a fabrication process based on conventional silicon microfabrication and sacrificial core layers, waveguides of varying widths and lengths with volumes covering the pico- to nanoliter range were fabricated. We observe confined mode propagation, measure waveguide losses of 2.4/cm, and demonstrate that the waveguides possess tailorable wavelength selectivity. The potential for highly integrated, sensitive devices based on these …


Wind Speed Effect On L-Band Brightness Temperature Inferred From Eurostarrs And Wise 2001 Field Experiments, David G. Long, Jacqueline Etcheto, Emmanuel P. Dinnat, Jacqueline Boutin, Adriano Camps, Stephanie Contardo, J. Wesson, Jordi Font, J. Miller Oct 2004

Wind Speed Effect On L-Band Brightness Temperature Inferred From Eurostarrs And Wise 2001 Field Experiments, David G. Long, Jacqueline Etcheto, Emmanuel P. Dinnat, Jacqueline Boutin, Adriano Camps, Stephanie Contardo, J. Wesson, Jordi Font, J. Miller

Faculty Publications

The results from two field experiments in the Mediterranean Sea are used to study the wind speed dependence of brightness temperature at L-band. During the EuroSTARRS airborne experiment, an L-band radiometer made measurements across a large wind speed gradient, enabling us to study this dependence at high wind speed. We compare our results with a two-scale emissivity model using several representations of the sea state spectrum. While the results are encouraging, unfortunately the accuracy of the measurements does not permit us to distinguish between the so-called twice Durden and Vesecky spectrum and the Elfouhaily spectrum above 7 m X s-1. …


Semi-Autonomous Human-Uav Interfaces For Fixed-Wing Mini-Uavs, Randal Beard, Michael A. Goodrich, Morgan Quigley Oct 2004

Semi-Autonomous Human-Uav Interfaces For Fixed-Wing Mini-Uavs, Randal Beard, Michael A. Goodrich, Morgan Quigley

Faculty Publications

Sponsorship: DARPA, AFOSR. We present several human-robot interfaces that support real-time control of a small semi-autonomous UAV. These interfaces are designed for searching tasks and other missions that typically do not have a precise predetermined flight plan. We present a detailed analysis of a PDA interface and describe how our other interfaces relate to this analysis. We then offer quantative and qualitative performance comparisons of the interfaces, as well as an analysis of their possible real-world applications.


High Order Convergence With A Low Order Discretization Of The 2d Mfie, Clayton P. Davis, Karl F. Warnick Sep 2004

High Order Convergence With A Low Order Discretization Of The 2d Mfie, Clayton P. Davis, Karl F. Warnick

Faculty Publications

Moment method solutions to the MFIE are often less accurate for a given grid than corresponding solutions to the EFIE. We propose that the cause of this observation is the identity operator in the MFIE and show how regularizing the identity increases the convergence rate of the discretized 2D MFIE by three orders.


Electromagnetic Bias At Off-Nadir Incidence Angles, David V. Arnold, Floyd W. Millet, Karl F. Warnick Sep 2004

Electromagnetic Bias At Off-Nadir Incidence Angles, David V. Arnold, Floyd W. Millet, Karl F. Warnick

Faculty Publications

Sponsorship: JPL Subcontract 961461. Nadir and off-nadir measurements of electromagnetic (EM) bias measurements are presented and compared with an off-nadir bias model. Measurements of the bias were made during the BYU Off-Nadir Experiment (Y-ONE) in the months of March and April, 2003. Using radar measurements of the surface and backscattered power, the EM bias was computed at angles from nadir to 17º degrees. Simultaneous surface measurements from a laser rangefinder provide accurate measurements of the long wave surface parameters. An off-nadir bias model incorporating the effects of hydrodynamic modulation of short waves and tilt modulation of long waves is developed …


Evaluation Of Power Costs In Applying Tmr To Fpga Designs, Nathaniel Rollins, Michael J. Wirthlin, Paul S. Graham Sep 2004

Evaluation Of Power Costs In Applying Tmr To Fpga Designs, Nathaniel Rollins, Michael J. Wirthlin, Paul S. Graham

Faculty Publications

Sponsorship: Los Alamos National Laboratory. Triple modular redundancy (TMR) is a technique commonly used to mitigate against design failures caused by single event upsets (SEUs). The SEU immunity that TMR provides comes at the cost of increased design area and decreased speed. Additionally, the cost of increased power due to TMR must be considered. This paper evaluates the power costs of TMR and validates the evaluations with actual measurements. Sensitivity to design placement is another important part of this study. Power consumption costs due to TMR are also evaluated in different FPGA architectures. This study shows that power consumption rises …


Detection Of Configuration Memory Upsets Causing Persistent Errors In Sram-Based Fpgas, D. Eric Johnson, Keith S. Morgan, Michael J. Wirthlin, Michael P. Caffrey, Paul S. Graham Sep 2004

Detection Of Configuration Memory Upsets Causing Persistent Errors In Sram-Based Fpgas, D. Eric Johnson, Keith S. Morgan, Michael J. Wirthlin, Michael P. Caffrey, Paul S. Graham

Faculty Publications

Sponsorship: Los Alamos National Laboratory. FPGA designers are becoming increasingly aware of fault tolerance issues in modern FPGA designs, especially designs destined for a radiation environment. We classify errors due to upsets within the configuration bitstream into two categories; namely, persistent and non-persistent. Persistent errors generally cannot be tolerated. However, non-persistent errors can be tolerated in certain types of designs as long as they are properly accounted for. We discuss situations in which non-persistent errors are acceptable, and describe a technique for the detection of upsets causing persistent errors within the configuration memor of an SRAM-based FPGA.


Optimal Sampling For Single-Tone Frequency Estimation, Travis E. Oliphant Aug 2004

Optimal Sampling For Single-Tone Frequency Estimation, Travis E. Oliphant

Faculty Publications

Single-tone frequency estimation from irregularly spaced samples is analyzed. The ratio of the Cramr-Rao Bound (CRB) for regularly spaced samples to the bound for irregularly spaced samples is presented and shown to be a globally convex function. It's minima and maxima over the domain of interest are identified which reveals that improvements in the CRB can be obtained using non-equally-spaced samples. For a given sampling density, the best estimates of frequency are obtained when half of the samples are taken at one end of the total available interval and the remaining samples are taken at the other end.


Noncontact Scanning Impedance Imaging In An Aqueous Solution, Aaron R. Hawkins, Hongze Liu, Travis E. Oliphant, Stephen M. Schultz Aug 2004

Noncontact Scanning Impedance Imaging In An Aqueous Solution, Aaron R. Hawkins, Hongze Liu, Travis E. Oliphant, Stephen M. Schultz

Faculty Publications

We present a method for imaging based on noncontact electrical impedance measurements and mechanical scanning. Measurement results are shown for an initial system based on this concept. An impedance probe design is presented, applicable to the test system. Line-scan data plots of high impedance contrast structures show a good fit to a theoretical physical model. Image resolutions on the order of 100 m are indicated for the initial system. Two-dimensional impedance images of biological tissue generated by this technique are shown.


High Efficiency 90 Degree Silica Waveguide Bend Using An Air Hole Photonic Crystal Region, J. Cai, S. Kim, Gregory P. Nordin, J. Jiang Aug 2004

High Efficiency 90 Degree Silica Waveguide Bend Using An Air Hole Photonic Crystal Region, J. Cai, S. Kim, Gregory P. Nordin, J. Jiang

Faculty Publications

We propose the hybrid integration of an air hole photonic crystal (PhC) structure with a high (0.75%) single-mode silica waveguide to achieve an ultracompact high efficiency 90 bend for transverse-magnetic polarized light. Diffraction from the periodic boundary between the PhC and silica waveguide regions is shown to seriously degrade the optical efficiency of the bend. A microgenetic algorithm ( GA) combined with a two-dimensional finite-difference time-domain method is used to modify the PhC and its boundary layer to suppress this diffraction which in turn maximizes bend efficiency. The final optimized structure has a 99.4% bend efficiency at a wavelength of …


3d Analysis Of Hybrid Photonic Crystal/Conventional Waveguide 90° Bend, J. Cai, S. Kim, Gregory P. Nordin, J. Jiang Jul 2004

3d Analysis Of Hybrid Photonic Crystal/Conventional Waveguide 90° Bend, J. Cai, S. Kim, Gregory P. Nordin, J. Jiang

Faculty Publications

We present a three-dimensional (3D) analysis of a hybrid photonic crystal conventional waveguide 90° bend proposed previously [Opt. Express 10, 1334 (2002)] as an ultracompact component for large-scale planar lightwave circuit integration. Both rigorous 3D finite-difference time-domain modeling and a simple perfect mirror model analysis were carried out for different Si post heights in the photonic crystal region. Results show that the bend efficiency increases rapidly with Si post height. For a post height of 6.5 µm, this structure yields a bend efficiency of 97.3% at a wavelength of 1.55 µm for 90° bends in 2 µm X 2 µm …


Cavity Enhancement Of The Magneto-Optic Kerr Effect For Optical Studies Of Magnetic Nanostructures, Aaron R. Hawkins, N. Qureshi, Holger Schmidt Jul 2004

Cavity Enhancement Of The Magneto-Optic Kerr Effect For Optical Studies Of Magnetic Nanostructures, Aaron R. Hawkins, N. Qureshi, Holger Schmidt

Faculty Publications

We present a study of cavity enhancement of the magneto-optic Kerr effect using dielectric multilayers in order to facilitate optical studies of individual single-domain nanomagnets. We develop a transfer matrix theory to analyze Kerr rotation from an arbitrary number of possibly lossy dielectric layers. The combination of one lossless and one thin metallic layer is found to be most favorable for studying individual nanomagnets, providing the best tradeoff between signal enhancement and spatial resolution. Accounting for the microscopic surface structure, we find good agreement between theory and experiment. Using this technique, we demonstrate Kerr enhancements by a factor of more …


Simultaneous Wind And Rain Retrieval Using Seawinds Data, David G. Long, David W. Draper Jul 2004

Simultaneous Wind And Rain Retrieval Using Seawinds Data, David G. Long, David W. Draper

Faculty Publications

The SeaWinds scatterometers onboard the QuikSCAT and the Advanced Earth Observing Satellite 2 measure ocean winds on a global scale via the relationship between the normalized radar backscattering cross section of the ocean and the vector wind. The current wind retrieval method ignores scattering and attenuation of ocean rain, which alter backscatter measurements and corrupt retrieved winds. Using a simple rain backscatter and attenuation model, two methods of improving wind estimation in the presence of rain are evaluated. First, if no suitable prior knowledge of the rain rate is available, a maximum-likelihood estimation technique is used to simultaneously retrieve the …


Automatic Detection And Validity Of The Sea-Ice Edge: An Application Of Enhanced-Resolution Quikscat/Seawinds Data, David G. Long, Jorg Haarpaintner, Rasmus T. Tonboe, Michael L. Van Woert Jul 2004

Automatic Detection And Validity Of The Sea-Ice Edge: An Application Of Enhanced-Resolution Quikscat/Seawinds Data, David G. Long, Jorg Haarpaintner, Rasmus T. Tonboe, Michael L. Van Woert

Faculty Publications

Sea-ice edge detection is an essential task at the different national ice services to secure navigation in ice-covered seas. Comparison between the Remund and Long ice mask image from enhanced-resolution QuikScat/SeaWinds (QS) products and the analyzed ice edge from high-resolution RADARSAT synthetic aperture radar has shown that the automatically determined QS ice mask underestimates the Arctic ice extent. QS data was statistically analyzed by colocating the data with ice charts around Greenland and with the National Aeronautics and Space Administration Team's Special Sensor Microwave/Imager (SSM/I) ice concentration algorithm over the whole Arctic region. All variables, i.e., the backscatter in vertical …


Synergetic Control Of Power Converters For Pulse Current Charging Of Advanced Batteries From A Fuel Cell Power Source, Zhenhua Jiang, Roger A. Dougal Jul 2004

Synergetic Control Of Power Converters For Pulse Current Charging Of Advanced Batteries From A Fuel Cell Power Source, Zhenhua Jiang, Roger A. Dougal

Faculty Publications

This paper presents a synergetic controller for pulse current charging of advanced batteries from a fuel cell power source. Pulse current charging protocol that has been shown to have many advantages over the traditional constant current/constant voltage protocol is applied in a fuel cell powered battery-charging station to reduce the total charging time. Strong nonlinearity and dynamics exist in such systems. In this paper, the synergetic control approach is applied to regulate the buck converters that control the pulse charging currents to the many batteries. A practical synergetic controller to coordinate pulse current charging of the battery is synthesized and …


Unmanned Air Vehicle Testbed For Cooperative Control Experiments, Timothy W. Mclain, Randal W. Beard Jul 2004

Unmanned Air Vehicle Testbed For Cooperative Control Experiments, Timothy W. Mclain, Randal W. Beard

Faculty Publications

This paper describes the MAGICC Lab unmanned air vehicle (UAV) testbed at Brigham Young University. Motivation for pursuing experimental research with UAVs is given as well as a historical perspective of the UAV testbed development. Lessons learned through the development and use of the testbed over the past several years are summarized.


Assessing The Quality Of Seawinds Rain Measurements, David G. Long, David W. Draper Jul 2004

Assessing The Quality Of Seawinds Rain Measurements, David G. Long, David W. Draper

Faculty Publications

While SeaWinds was designed to measure ocean winds, it can also measure rain over the ocean. SeaWinds on QuikSCAT active measurements of integrated columnar rain rate obtained via simultaneous wind/rain retrieval are evaluated via Monte Carlo simulation and the Cramer-Rao lower bound on estimate accuracy. Although sufficiently accurate in many conditions, the simultaneous wind/rain retrieval method used with SeaWinds on QuikSCAT data is ill-conditioned for certain wind directions and measurement geometries, sometimes yielding spurious rain rates in zero-rain conditions. To assess the validity of SeaWinds-derived rain rates, a simple empirically based rain thresholding scheme is presented, derived from simulated data. …


Mutual Coupling In Mimo Wireless Systems: A Rigorous Network Theory Analysis, Michael A. Jensen, Jon W. Wallace Jul 2004

Mutual Coupling In Mimo Wireless Systems: A Rigorous Network Theory Analysis, Michael A. Jensen, Jon W. Wallace

Faculty Publications

A new framework for the analysis of multiple-input multiple-output (MIMO) wireless systems is introduced to account for mutual coupling effects in the antenna arrays. The multiport interactions at transmit and receive are characterized by representing the channel using a scattering parameter matrix. A new power constraint that limits the average radiated power is also introduced. The capacity of the MIMO system with mutual coupling is defined as the maximum mutual information of the transmit and receive vectors over all possible transmit signaling and receive loading. Full-wave electromagnetic antenna simulations combined with a simple path-based channel model are used to demonstrate …


Using Open-Ended Design Projects In An Open Lab To Teach Lifelong Learning Skills In An Ic Design Course, David Parent Jun 2004

Using Open-Ended Design Projects In An Open Lab To Teach Lifelong Learning Skills In An Ic Design Course, David Parent

Faculty Publications

The ABET criterion, recognition for and the ability to engage in life-long learning, is one of the most important, given that we as engineering educators can not teach every concept that students will encounter over the span of a 20-40 year career as an engineer. We believe that one method to teach and verify this critical skill in the area of Integrated Circuit (IC) design, is to have the students learn IC CAD tools by following well crafted tutorials and then complete open-ended IC design projects in an open lab. The project is supposed to be a full custom design …


Autostereoscopic 3d Display Based On Micromirror Array, Gregory P. Nordin, C. H. Ahn, H. J. Cho, S. T. Kowel, J. H. Kulick, J. Yan Jun 2004

Autostereoscopic 3d Display Based On Micromirror Array, Gregory P. Nordin, C. H. Ahn, H. J. Cho, S. T. Kowel, J. H. Kulick, J. Yan

Faculty Publications

A novel approach for three-dimensional (3-D) display systems implemented with a micromirror array was proposed, designed, realized, and tested. The major advantages of this approach include the following: 1 micromirrors are reflective and hence achromatic (panchromatic), 2 a wide variety of displays can be used as image sources, and 3 time multiplexing can be introduced on top of space multiplexing to optimize the viewing zone arrangements. A two-view (left and right) 3-D autostereoscopic display system was first constructed. Left- and right-eye views in the forms of both still and motion 3-D scenes were displayed, and viewers were able to fuse …


Correlation And Covariance Of Satellite Scatterometer Measurements, David G. Long, Peter K. Yoho Jun 2004

Correlation And Covariance Of Satellite Scatterometer Measurements, David G. Long, Peter K. Yoho

Faculty Publications

Capable scatterometer designs have been developed that seek improved resolution through the use of higher pulse sampling rates. The high sampling rates bring the traditional scatterometry assumption of measurement independence into question. This paper uses fundamental scattering theory to derive general expressions for correlation and covariance between scatterometer measurements and provides practical analysis using current and future instruments as examples. The paper derives expressions for the measurement variance parameter Kp when measurement correlation due to Rayleigh fading effects is present and relates Kp to the statistics of multiple pulse measurements. A function of the transmit signal modulation and receive processing, …


Response To Comments On “Passivity-Based Control Of Saturated Induction Motors”, Levent U. Gökdere, Marwan A. Simaan, Charles W. Brice Jun 2004

Response To Comments On “Passivity-Based Control Of Saturated Induction Motors”, Levent U. Gökdere, Marwan A. Simaan, Charles W. Brice

Faculty Publications

Contrary to the claims made in the comments to our paper, the passivity-based controller developed for induction motors has already been tested on the same demanding trajectories used for the input–output linearization controller. The experimental results show that the passivitybased controller provides closer tracking of the same mechanical trajectory, when compared with the input–output linearization controller.


Simulation Of Hot Electron And Quantum Effects In Algan/Gan Heterostructure Field Effect Transistors, N. Braga, R. Mickevicius, R. Gaska, H. Xu, M. S. Shur, M. Asif Khan, Grigory Simin, J. Yang Jun 2004

Simulation Of Hot Electron And Quantum Effects In Algan/Gan Heterostructure Field Effect Transistors, N. Braga, R. Mickevicius, R. Gaska, H. Xu, M. S. Shur, M. Asif Khan, Grigory Simin, J. Yang

Faculty Publications

We report on simulations of electrical characteristics of AlGaN/(InGaN)/GaN heterostructurefield effect transistors with quantum and hot electroneffects taken into account. Polarization charges lead to quantum confinement of electrons in the channel and to the formation of two-dimensional electron gas. The electron quantization leads to the spread of the electronwave function into the barrier and bulk but does not have significant impact on dc electrical characteristics.Hot electrons play an important part in the charge transport by spilling over into the bulk GaN where they are captured by traps. This leads to negative differential conductivity, which is also observed experimentally. The simulation …


Linear Voltage To Current Conversion Using Submicron Cmos Devices, David J. Comer, Donald Comer, Aaron Shreeve May 2004

Linear Voltage To Current Conversion Using Submicron Cmos Devices, David J. Comer, Donald Comer, Aaron Shreeve

Faculty Publications

Sponsorship: AMIS--American Micro-Systems, Inc. This paper investigates the linearity of submicron gate length CMOS devices and their behavior in open loop voltage to current (V-I) converters. Methods are developed to estimate the deviation from linearity in V-I converters due to short-channel effects. Using these methods, a converter is designed and fabricated on a 0.35 micron process. The measured deviation from linearity is less than 1% and the simulated bandwidth is 1 GHz.


Control Strategies For Active Power Sharing In A Fuel-Cell-Powered Battery-Charging Station, Zhenhua Jiang, Roger A. Dougal May 2004

Control Strategies For Active Power Sharing In A Fuel-Cell-Powered Battery-Charging Station, Zhenhua Jiang, Roger A. Dougal

Faculty Publications

This paper presents an effective system design for a fuel-cell-powered battery-charging station and three control strategies for active power sharing among the batteries. This battery-charging station allows multiple batteries to be simultaneously charged. Three control strategies were investigated to coordinate the active power distribution among the battery-charging branches. The baseline control strategy Was equal rate charging. Two advanced control strategies, proportional rate charging and pulse current charging, were compared to the baseline strategy. These control strategies were realized in MaTLaB/Simulink, and the current and voltage regulations were implemented using the classical proportional-integral control approach. The system simulation was conducted in …


On Noncoherent Detection Of Cpm, Michael D. Rice, Erik Perrins Apr 2004

On Noncoherent Detection Of Cpm, Michael D. Rice, Erik Perrins

Faculty Publications

This work was supported under a grant from the Air Force ARTM program (Contracts F04700-02-P-0080 and F04700-02-P-0081). It is well understood that the performance of noncoherent receivers with multi-symbol observation intervals approaches that of coherent receivers as the observation interval grows arbitrarily large. However, since complexity also grows exponentially with observation length, there are practical limits to this approach. In this paper we present a noncoherent receiver for continuous phase modulation (CPM) whose structure is a hybrid between existing coherent and noncoherent receiver architectures. The presentation is given in the most general M-ary multi-h terms, with some emphasis on the …